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A seven-mode truncation system of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is considered. Its stationary solutions and stability are presented; the existence of the attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through instability of invariant tori, is simulated numerically by computers with the changing of Reynolds number. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system, some basic dynamical behaviors of the new chaos system are revealed.

In recent years much attention has been devoted to the study of simple differential or difference equations, which although deterministic, exhibit a transition as some parameters go through certain values to a chaos behavior. The equations which are studied often arise in a natural way as simplified models in fluid dynamics and in ecology. The best known examples are perhaps the models of [

Consider the incompressible Navier-Stokes equations:

on the torus

We expanded

where

where

We take as the set of vectors

Taking the force acting on the mode

of Calculation we obtain the following system

In this section we discuss the stationary solution and their stability properties of the system (2.8). Let

setting

(a) For

stable for

(b) For

where

(c) For

In the following we prove the existence of attractor of the system (8).

By calculating

accordingly,

letting

as a result,

, then

From above we have

If

When the system is a global stability,its orbits contract into a domain called the trapping region. Therefore, if the existence of the trapping region is proved, the system has the global stability, though the stationary solutions are unstable. We construct a following Liapunov function of the system (8)

Setting

Obviously

From the Liapunov theory we know that the orbits out of system (8) will enter E. Namely E is the trapping region of the Equations (8). Though the stationary solutions

With the increasing of Reynolds number

1) At

2) When

3) At

4) With the increasing of the Reynolds number r, a strong hysteresis phenomenon(i.e., coexistence of stable attractors) appears, in some intervals hysteresis takes place between closed orbits and tori (Figures 9-19).

5) At

6)

7) Figures 28-30 show Poincare section, return map and power spectrum of the system (2.8) when

8) For

In this work we have reported the results of our theoretical and numerical investigation on a model of seven nonlinear ordinary differential equations. Such a model, obtained by a suitable seven-mode truncation of the Navier-Stokes equations for an incompressible fluid on a torus, exhibits a very varied phenomenology, with an

interesting sequence of bifurcations. From numerical results we present four different and independent stories describing the complete phenomenology of the model. The first story consists of a sequence of a bifurcations very similar to the one found by Curry in [

For all the values of the Reynolds number r larger than 158.631, when no stable periodic orbits or tori are present any more, the model exhibits a turbulent behavior. In fact any randomly chosen point describes trajectories which appear to be completely random and sensitively dependent on initial conditions. Since all the numerical investigations carried up to

We thank the editor and the referee for their comments. Research work is funded by the funds for education department of Liaoning Province (L2013248) and science and technology funds of Jinzhou city (13A1D32).